QUANTIZED SYSTEMS AND CONTROL

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QUANTIZED SYSTEMS AND CONTROL
Daniel Liberzon
Coordinated Science Laboratory and Dept. of
Electrical Computer Eng., Univ. of Illinois at
Urbana-Champaign
DISC HS, June 2003
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(No Transcript)
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REASONS for SWITCHING

• Nature of the control problem

• Sensor or actuator limitations

• Large modeling uncertainty

• Combinations of the above

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REASONS for SWITCHING

• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above

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CONSTRAINED CONTROL
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LIMITED INFORMATION SCENARIO
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MOTIVATION

• Limited communication capacity
• many systems sharing network cable or wireless
  medium
• microsystems with many sensors/actuators on one
  chip

• Need to minimize information transmission
  (security)

• Event-driven actuators
• PWM amplifier
• manual car transmission
• stepping motor

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QUANTIZED CONTROL ARCHITECTURES
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QUANTIZER GEOMETRY
Dynamics change at boundaries gt hybrid
closed-loop system
Chattering on the boundaries is possible (sliding
mode)
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QUANTIZATION ERROR and RANGE
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EXAMPLES of QUANTIZERS

• A/D conversion

• Coding and decoding

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OBSTRUCTION to STABILIZATION
Assume fixed
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BASIC QUESTIONS

• What can we say about a given quantized system?

• How can we design the best quantizer for
  stability?

• What can we do with very coarse quantization?

• What are the difficulties for nonlinear systems?

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BASIC QUESTIONS

• What can we say about a given quantized system?

• How can we design the best quantizer for
  stability?

• What can we do with very coarse quantization?

• What are the difficulties for nonlinear systems?

• What are the difficulties for nonlinear systems?

• What are the difficulties for nonlinear systems?

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STATE QUANTIZATION LINEAR SYSTEMS
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LINEAR SYSTEMS (continued)
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NONLINEAR SYSTEMS
For linear systems, we saw that if
gives then
automatically gives
when
This is robustness to measurement errors
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SUMMARY PERTURBATION APPROACH
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INPUT QUANTIZATION
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OUTPUT QUANTIZATION
Analysis same as before (need a bound on
initial state)
Can also treat input and state/output
quantization together
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BASIC QUESTIONS

• What can we say about a given quantized system?

• How can we design the best quantizer for
  stability?

• What can we do with very coarse quantization?

• What are the difficulties for nonlinear systems?

• What are the difficulties for nonlinear systems?

• What are the difficulties for nonlinear systems?

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LOCATIONAL OPTIMIZATION NAIVE APPROACH
Compare mailboxes in a city, cellular base
stations in a region
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MULTICENTER PROBLEM

This is the center of enclosing sphere of
smallest radius
iterate
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LOCATIONAL OPTIMIZATION REFINED APPROACH
Only applicable to linear systems
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WEIGHTED MULTICENTER PROBLEM
on not containing 0 (annulus)
Lloyd algorithm as before
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DYNAMIC QUANTIZATION IDEA
Temperature sensor can adjust threshold
settings Digital camera can zoom in and
out Encoder can change the coding mechanism
After ultimate bound is achieved, recompute
partition for smaller region
Zoom out to overcome saturation
Can recover global asymptotic stability
(also applies to input and output quantization)
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DYNAMIC QUANTIZATION DETAILS
(More realistic, easier to design and analyze,
robust to time delays)
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BASIC QUESTIONS

• What can we say about a given quantized system?

• How can we design the best quantizer for
  stability?

• What can we do with very coarse quantization?

• What are the difficulties for nonlinear systems?

• What are the difficulties for nonlinear systems?

• What are the difficulties for nonlinear systems?

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ACTIVE PROBING for INFORMATION
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LINEAR SYSTEMS
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LINEAR SYSTEMS
Example

• is divided by 3 at the sampling time

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LINEAR SYSTEMS (continued)
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NONLINEAR SYSTEMS
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NONLINEAR SYSTEMS

• is divided by 3 at the sampling time

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NONLINEAR SYSTEMS (continued)
The norm

• grows at most by the factor in
  one period

• is divided by 3 at each sampling time

Need ISS w.r.t. measurement errors!
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RESEARCH DIRECTIONS
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REFERENCES
Brockett L, 2000 (IEEE TAC) Bullo L, 2003
(submitted)